Problem 44 - University Physics with Modern Physics

Question

In the twin paradox example, Alice boards a spaceship that flies to a space station 3.25 light-years away and then returns with a speed of \(0.650 c .\) This can be viewed in terms of Alice's reference frame. a) Show that Alice must travel with a speed of \(0.914 c\) to establish a relative speed of \(0.650 c\) with respect to Earth when Alice is returning back to Earth. b) Calculate the time duration for Alice's return flight toward Earth with the aforementioned speed.

Step-by-Step Solution

Answer

Answer: Alice must travel at a speed of 0.914c to have a relative velocity of 0.65c with respect to Earth on her return trip. Her return flight will take approximately 2.14 years.

Step 1: Finding the relative velocity between Earth and Alice on her return

To find Alice's required speed for having a relative velocity of 0.65c with respect to Earth, we need to use the relativistic velocity addition formula: \(u' = \frac{u+v}{1 + \frac{uv}{c^2}}\) Here, \(u'\) is the relative velocity, \(u\) is Earth's velocity, \(v\) is Alice's velocity, and \(c\) is the speed of light. In this case, Alice's spaceship is moving in the opposite direction when returning (\(-0.650c\)). So, we have: \(u' = 0.650c, u = -0.650c\), and we need to find \(v\).

Step 2: Using the relativistic velocity addition formula

Plugging the values from Step 1 into the relativistic velocity addition formula, we get: \(0.650c = \frac{v - 0.650c}{1 - \frac{v(-0.650c)}{c^2}}\) Next, we solve this equation for \(v\): \(0.650 = \frac{v - 0.650}{1 + 0.650v} \Rightarrow\) \(0.650 + 0.650^2v = v - 0.650 \Rightarrow\) \(v(1 - 0.650^2) = 1.3 \Rightarrow\) \(v = \frac{1.3}{1 - 0.650^2} = 0.914\) So, Alice must travel with a speed of \(0.914c\) to have a relative speed of \(0.650c\) with respect to Earth when returning.

Step 3: Calculating the time for Alice's return flight using time dilation formula

To calculate the time duration of Alice's return flight, we first need to find the distance she travels in her reference frame. We can calculate this by using the Lorentz length contraction formula: \(L' = L \sqrt{1 - \frac{v^2}{c^2}}\) Here, \(L = 3.25\,\text{light-years}\) is the distance in Earth's frame, and \(L'\) is the distance in Alice's frame. Plugging the values into the length contraction formula, we get: \(L' = 3.25\, \sqrt{1 - \frac{(0.914c)^2}{c^2}}\) \(L' = 3.25\, \sqrt{1 - 0.914^2} \approx 1.96\,\text{light-years}\) Now we can use the time dilation formula to find the time duration of Alice's return flight: \(\Delta t' = \frac{\Delta t}{\sqrt{1 - \frac{v^2}{c^2}}}\) Here, \(\Delta t'\) is the time duration in Alice's frame, and \(\Delta t\) is the time duration in Earth's frame. We need to find \(\Delta t'\). Knowing the distance and velocity in Alice's frame, we can find the time duration as: \(\Delta t' = \frac{L'}{v} = \frac{1.96\,\text{light-years}}{0.914c}\) \(\Delta t' \approx 2.14\, \text{years}\) So, the time duration of Alice's return flight with the required speed is approximately 2.14 years.

Key Concepts

Relativistic Velocity Addition Formula

When Alice travels at a significant fraction of the speed of light, simply adding her spaceship's speed to Earth's would not provide an accurate measure of her relative speed due to the effects of special relativity. This is where the relativistic velocity addition formula comes into play. It corrects for the distortions that arise at high speeds. The formula is:
\[\begin{equation}u = \frac{u + v}{1 + \frac{uv}{c^2}}\tag{1}\text{where:}\end{equation}\]

• \begin{math}u\text{ is the resultant relative velocity.}\text{}
• \begin{math}u\text{ and }v\text{ are the velocities of the two moving objects in any frame.}
• \begin{math}c\text{ is the speed of light in a vacuum.}\text{}

In Alice's case, applying this formula helps determine the speed she needs to attain a relative velocity of \(0.650 c\) with respect to Earth during her return journey. This formula shows us that velocities at relativistic speeds don't simply add together like they do at everyday speeds we're used to.

Time Dilation

An astonishing consequence of traveling at relativistic speeds, as shown in Alice's journey, is time dilation. This effect predicts that time moves slower for an observer in motion, compared to an observer at rest. Albert Einstein's special theory of relativity provides the quantitative tool to calculate this temporal discrepancy with the time dilation formula:
\[\begin{equation} \begin{split} \tau = \frac{t}{\begin{math}\begin{vmatrix}1 - \frac{v^2}{c^2}\text{}

\begin{math}\begin{vmatrix}\tau\text{is the time experienced by the moving observer (Alice),}\text{}

t\text{is the proper time for the stationary observer (Earth),}\text{}

v\text{is the velocity of the moving observer, and} c^2\text{is the square of the speed of light.}\text{}

For Alice traveling back to Earth, the time dilation formula allows us to determine that she will experience less time than her twin who stays behind on Earth. This time difference is not due to the mechanical workings of clocks but rather to the fundamental nature of time itself as affected by motion at high speeds.

Lorentz Length Contraction

Along with time and velocity, distance also behaves in non-intuitive ways at relativistic speeds. This is manifest in the phenomenon known as Lorentz length contraction. Objects moving relative to an observer will appear shorter along the direction of motion, proportional to their speed:
\[\begin{equation}L' = L\begin{math}\begin{vmatrix}1 - \frac{v^2}{c^2}\text{}

\begin{math}L'\end{math}\text{is the contracted length as observed in the moving reference frame (Alice's spaceship),} \end{math}\text{}

L\text{is the proper length as measured in the rest frame (Earth),} \begin{math}\text{}

v\text{is the velocity of the moving object, and}\text{}

\begin{math} c^2\text{is the square of the speed of light.}\text{}

In the twin paradox that Alice undergoes, the distance to the target destination appears shorter in her own reference frame than it would appear to someone stationary relative to the space itself, like her twin on Earth. Lorentz contraction is a central aspect of special relativity that enforces the limit that no information or matter can travel faster than the speed of light.

Special Relativity

The unity of the concepts described above—relativistic velocity addition, time dilation, and Lorentz length contraction—are encompassed within Einstein's theory of special relativity. This theory fundamentally changed our understanding of space, time, and motion by describing how these are perceived differently by observers in different inertial frames of reference, especially at speeds comparable to the speed of light. It is a cornerstone that reconciled the laws of motion with the constancy of the speed of light.Special relativity can be summarized by two postulates:

• The laws of physics are invariant (identical) in all inertial frames of reference (those moving at constant velocity with respect to each other).
• The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or observer.

These postulates lead to the conclusion that the measurements of time intervals, spatial distances, and velocities are not absolute but depend on the relative motion of observers. The twin paradox example offers a vivid illustration of these effects, with Alice's incredible journey challenging our everyday experiences of time and space.